This page includes the script of the video. Sentences in italic are those presented in the clip.

Opening credits

Question about Thymio

Is Thymio an artist?

General concept presentation

Definition

For starters, let's try to define what a fractal figure is, or simply a "fractal". It is a geometrical form whose pattern repeats itself indefinitely. This can be observed at every scale.

Koch Curve

To understand how such a figure works, let's starting by drawing a line.

We divide this line into 3 equal pieces and then draw an equilateral triangle on the centre part. We can now remove this centre part, which has been replaced by 2 lines of the same size.

We continue our drawing by repeating the same steps on each new part. We could continue doing this indefinitely, but we will stop here. This figure is called the Koch Curve.

If we observe one side of our first iteration, we can see that the figure is identical to our whole figure. And if we take a small part of that figure, we can see that again, it is identical to the whole! This is what we call a fractal.

Dimension

To strictly describe a fractal, we must observe its dimension.

Different kinds of dimensions exist: Euclidean, topologic and fractal. In our case, only the fractal dimension is interesting. It is defined by d = log n / log m, where n is the number of parts obtained after iteration and m is the homothetic ratio (in how many parts we divide the initial part).

For the Koch curve, we divide our segment into 3 equal parts (so m=3) and we obtain 4 segments that are each 1/3 of the initial length (so n=4). This gives us a dimension d = ln 4/ln 3 = 1.26.

To see if our Koch curve is a fractal, its fractal dimension must either be greater than its topological dimension, or not a whole number. In our case this is verified, so our Koch curve is a fractal!

New definition

We can now give a more rigorously defined definition of a fractal: it is an object which:

is too irregular to be defined by the usual geometric vocabulary

is self-similar (which means that each part of the object resembles its whole)

has a non-whole dimension, or one that is greater than its topological dimension

What is it used for?

Now we know what a fractal is but… what is it used for? Many applications exist, but I will only cite a few.

First of all, we can make beautiful figures like this *show a picture* or this *show a picture*. But it also gives us a way of describing physical phenomena such as a fluid's turbulent behaviour or galaxies' organisation in space, or even the geometrical shape of a romanesco broccoli! *photo of a romanesco broccoli*

Fractals are also a way of compressing images with a constant quality for every zoom.

Concept presentation with Thymio

Now let's use Thymio to draw a fractal composed of a big circle with in it three smaller circles, which each hold three smaller circles. We could repeat this an infinite number of times.